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Authors
Chen, Han Fu

Digital Object Identifier (DOI)
10.3182/20090706-3-FR-2004.00069

Page Numbers:
420-425

Index Terms
Recursive Identification; Nonlinear System Identification

Abstract
A unified approach to recursively identifying ARMAX systems, Hammerstein systems, Wiener systems, and nonlinear ARX systems is presented, by which the problem is solved by two steps. First, the task of identification is transformed to a root-seeking problem by selecting a regression function, whose roots coincide with the estimated parameters, and by forming an available value called ``observation" at each time. However, the resulting root-seeking problem is hardly to be solved by the classical Robbins-Monro (RM) algorithm. Instead, the stochastic approximation (SA) algorithm with expanding truncations (SAAWET) and its general convergence theorem (GCT) serve as the main tool. The second step of the unified approach is to verify conditions required by GCT. Reasonable conditions are respectively given for each system mentioned above, under which the estimates given by the recursive algorithms are strongly consistent.

References

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\bf References

\rm \begin{description}
{%\scriptsize%mall
\item{[1]} E. W. Bai and D. Li, Convergence of the iterative Hammerstein
system identification algorithm, {\it IEEE Trans. Automat. Contr.},
Vol. 49, No. 11, pp. 1929--1940, 2004.

\item{[2]} E. W. Bai and Y. Liu, Recursive direct weight optimization in
nonlinear system identification: a minimal probability approach,
{\it IEEE Trans. Automat. Contr.}, Vol. 52, No. 7, pp. 1218-1231,
2007.

\item{[3]} E. W. Bai, R. Tempo, and Y. Liu, Identification of IIR nonlinear
systems without prior structural information, {\it IEEE Trans.
Automat. Contr.}, Vol. 52, No. 3, pp. 442-453, 2007.


\item{[4]} A. Benveniste, M. M\'{e}tivier, and P. Priouret,
Adaptive Algorithms and Stochastic Approximations, New York,
Springer Verlag, 1990.


\item{[5]} H. F. Chen, Stochastic Approximation and Its Applications, Kluwer,
Dordrecht,  2002.

\item {[6]} H. F. Chen, Almost surely convergence of iterative learning control
      for stochastic systems, {\it Science in China\,} (Series F), Vol. 46, No. 1, 2003, 69-79.

\item{[7]} H. F. Chen, Pathwise convergence of recursive identification algorithms for
Hammerstein systems, IEEE Trans. Automat. Contr., Vol. 49, No. 10,
pp.1641--1649, 2004.

\item{[8]} H. F. Chen, Recursive identification for Wiener model with
discontinuous piece-wise linear function, {\it IEEE Trans. Automat.
Contr.}, Vol. 51, No. 3, pp. 390--400, 2006.

\item {[9]} H. F. Chen, Adaptive regulator for Hammerstein and Wiener
      systems with noisy observations, {\it IEEE Trans. Autom. Control}, Vol. 52, No. 4, 703-709, 2007.

\item {[10]} H. F. Chen, Recursive identification for multivariate errors-in-variables systems,
      {\it Automatica}, Vol. 43, No. 7, 1234-1242, 2007.

\item {[11]} H. F. Chen, Recursive identification for ARMAX systems,
Proceedings of the 27th Chinese Control Conference, July 16-18,
2008, Kunming, China, Vol. 3, 199-203.

\item {[12]} H. F. Chen and H. T. Fang, Output tracking for nonlinear
      stochastic systems by iterative learning control, {\it IEEE Trans. Autom. Control}, Vol. 49, No. 4, 583-588, 2004.

\item {[13]} H. F. Chen and L. Guo, Identification and Stochastic
Adaptive Control, Birkh\"{a}user, Boston, 1991.

\item {[14]} H. F. Chen and G. Yin, Asymptotic properties of sign
      algorithms for adaptive filtering, {\it IEEE
      Trans. Autom. Control}, Vol. 48, No. 9, 2003, 1545-1556.

\item{[15]} H. F. Chen and Y.M. Zhu, Stochastic approximation procedures  with
randomly varying truncations,  Scientia Sinica (Series A), Vol.29,
No.9, 1986, 914-926.

\item{[16]} W. Greblicki, Stochastic approximation in nonparametric
identification of Hammerstein systems, {\it IEEE Trans. Automat.
Contr.}, Vol. 47,  No. 11, pp. 1800-1810, 2002.


\item{[17]} E. J. Hannan, The identification of vector mixed autoregressive-moving average
systems, {\it Biometrica,} 56, 1969, 223-225.

\item{[18]} E. J. Hannan, The estimation of ARMA models, {\it The Annals of
Statistics,} Vol.~3, No.~4, 1975, 975--981.


\item{[19]}  X. L. Hu and H. F. Chen, Strong consistency of recursive
      identification for Wiener systems, {\it Automatica}, Vol. 41, No. 11, 1905-1916, 2005.

\item{[20]} X. L. Hu and H. F. Chen, Identification for Wiener Systems with RTF Subsystems,
      {\it European Journal of Control,} vol. 6, 581-594, 2006.

\item{[21]} Y. Q. Huang, H. F. Chen, and H. T. Fang, Identification of Wiener Systems with Nonlinearity
      Being Piecewise-linear Function, {\it Science in China Series F: Information Science}, Vol. 51, No. 1, 1-12, 2008.

\item {[22]} Y.P. Jiang and H. T. Fang, Recursive Subspace Identification for Closed-loop Systems, Proceeding of 26th Chinese Control Conference£¬2007£¬
      Zhangjiajie£¬ Hunan, Vol. 3,  pp. 287-291.


\item{[23]} H. J. Kushner and G. G. Yin, Stochastic Approximation
and Recursive Algorithms and Applications, Second Edition, Springer,
New York, 2003.

\item{[24]} T. L. Lai and C. Z. Wei, Least squares estimates in
stochastic regression models with application to identification and
control of dynamic systems, Ann. Stat., Vol. 10, No. 1, 154-166,
1983.

\item{[25]} L. Ljung, {\it System Identification: Theory for Users}, Upper
Saddle River, NJ: Prentice Hall, 2nd edition, 1999.

\item{[26]} L. Ljung, Perspectives on system identification, Plenary Papers,
Milestone Reports \& Selected Survey Papers, (Edited by M. J. Chung
and P. Misra), 17th IFAC World Congress, July 6-11, 2008, Seoul,
pp.47-59.

\item{[27]} L. Ljung and A. Vicino, editors, IEEE Trans. Automatic
      Control: Special Issue on Identification, Vol. AC-50, Oct.
      2005.

\item{[28]} H. Robbins and S. Monro, A stochastic approximation
method, Ann. Math. Statist., Vol. 22, 1951, 400-407.

\item{[29]}  Q. J. Song and H. F. Chen, Identification of  errors-in-variables systems with ARMA observation
      noises, {\it Systems and Control Letters},  Vol. 57, No. 5, 420-424, 2008.

\item {[30]} Q. J. Song and H. F. Chen, Identification for Wiener Systems With Internal
      Noise, {\it Journal of Systems Science \& Complexity,} Vol. 21, No. 3, 378-393, 2008.

\item{[31]} P. Stoica, T. McKelvey, and J. Mari, J.,  MA estimation in
polynomial time, IEEE Trans. Signal Processing, Vol. 48, No. 7,
2000, 1999-2012.

\item{[32]} J. V\"{o}r\"{o}s, Parameter identification of Wiener systems with
multisegment piecewise-linear nonlinearities, {\it Systems and
Control Letters,} Vol. 56, pp. 99-105, 2007.

\item{[33]}  W. X. Zhao and H. F. Chen, Recursive identification for Hammerstein system with ARX subsystem,
      {\it IEEE Trans. Autom. Control}, Vol. 51, No. 12, 1966-1974, 2006.

\item{[34]} W. X. Zhao and H. F. Chen, Recursive identification
for nonlinear ARX systems, submitted for publication.

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